arXiv subjects

Hoang Ly

Publications and source records attributed to Hoang Ly.

7 recordsLinked to original sources

arXiv2026

Majority-Logic Decoding of Binary Locally Recoverable Codes: A Probabilistic Analysis

Locally repairable codes (LRCs) were originally introduced to enable efficient recovery from erasures in distributed storage systems by accessing only a small number of other symbols. While their structural properties-such as bounds and constructions-have been extensively studied, the performance of LRCs under random erasures and errors has remained largely unexplored. In this work, we study the error- and erasure-correction performance of binary linear LRCs under majority-logic decoding (MLD). Focusing on LRCs with fixed locality and varying availability, we derive explicit upper bounds on the probability of decoding failure over the memoryless Binary Erasure Channel (BEC) and Binary Symmetric Channel (BSC). Our analysis characterizes the behavior of the bit-error rate (BER) and block-error rate (BLER) as functions of the locality and availability parameters. We show that, under mild growth conditions on the availability, the block decoding failure probability vanishes asymptotically, and that majority-logic decoding can successfully correct virtually all of error and erasure patterns of weight linear in the blocklength. The results reveal a substantial gap between worst-case guarantees and typical performance under stochastic channel models.

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arXiv2025

Universal Maximum Likelihood (List) Decoding via Fast Vector-Matrix Multiplication

Maximum-likelihood (ML) decoding for arbitrary block codes remains fundamentally hard, with worst-case time complexity-measured by the total number of multiplications-being no better than straightforward exhaustive search, which requires $q^{k} n$ operations for an $[n,k]_q$ code. This paper introduces a simple, code-agnostic framework that reduces the worst-case complexity by a factor of $n$, down to $q^{k}$ operations, a highly desirable reduction in practice. The result holds for both linear and nonlinear block codes over general memoryless channels and under both hard-decision and soft-decision decoding. It naturally extends to intersymbol-interference (ISI) channels and ML list decoding with only a negligible increase in complexity. Our core insight is that, upon receipt of each sequence at the receiver, the conditional probability of that sequence for each codeword in the codebook (i.e., the \emph{likelihood}) can be expressed as the inner product of two carefully constructed vectors -- the first depending on the received sequence, and the second on that codeword itself. As a result, evaluating the likelihoods for all codewords in the codebook reduces to a single vector-matrix multiplication, and ML decoding (MLD) becomes the simple task of picking the maximum entry in the resulting vector. The only non-trivial cost lies in the vector-matrix product. However, our matrix construction allows the use of the Mailman algorithm to reduce this cost. This time reduction is achieved at the cost of high space complexity, requiring $\mathcal{O}(q^{k+1} n)$ space to store the pre-computed codebook matrix.

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arXiv2025

Optimum 1-Step Majority-Logic Decoding of Binary Reed-Muller Codes

The classical majority-logic decoder proposed by Reed for Reed-Muller codes RM(r, m) of order r and length 2^m, unfolds in r+1 sequential steps, decoding message symbols from highest to lowest degree. Several follow-up decoding algorithms reduced the number of steps, but for a limited set of parameters, or at the expense of reduced performance, or relying on the existence of some combinatorial structures. We show that any one-step majority-logic decoder-that is, a decoder performing all majority votes in one step simultaneously without sequential processing-can correct at most d_min/4 errors for all values of r and m, where d_min denotes the code's minimum distance. We then introduce a new hard-decision decoder that completes the decoding in a single step and attains this error-correction limit. It applies to all r and m, and can be viewed as a parallel realization of Reed's original algorithm, decoding all message symbols simultaneously. Remarkably, we also prove that the decoder is optimum in the erasure setting: it recovers the message from any erasure pattern of up to d_min-1 symbols-the theoretical limit. To our knowledge, this is the first 1-step decoder for RM codes that achieves both optimal erasure correction and the maximum one-step error correction capability.

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arXiv2025

Maximal Achievable Service Rates of Codes and Connections to Combinatorial Designs

We investigate the service-rate region (SRR) of distributed storage systems that employ linear codes. We focus on systems where each server stores one code symbol, and a user recovers a data symbol by accessing any of its recovery groups, subject to per-server capacity limits. The SRR--the convex polytope of simultaneously achievable request rates--captures system throughput and scalability. We first derive upper and lower bounds on the maximum request rate of each data object. These bounds hold for all linear codes and depend only on the number of parity checks orthogonal to a particular set of codeword coordinates associated with that object, i.e., the equations used in majority-logic decoding, and on code parameters. We then check the bound saturation for 1) all non-systematic codes whose SRRs are already known and 2) systematic codes. For the former, we prove the bounds are tight. For systematic codes, we show that the upper bound is achieved whenever the supports of minimum-weight dual codewords form a 2-design. As an application, we determine the exact per-object demand limits for binary Hamming codes. Our framework provides a new lens to address the SRR problem through combinatorial design theory.

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arXiv2025

Service Rate Regions of MDS Codes & Fractional Matchings in Quasi-uniform Hypergraphs

The service rate region (SRR) has emerged as a critical performance metric for distributed systems that store data redundantly. It measures the system's ability to serve multiple users concurrently. Mathematically, the SRR is a polytope in R^k where each dimension corresponds to the service request rate of one of the k data objects. This paper focuses on systems employing a class of Maximum Distance Separable (MDS) codes. For each code in the class, we characterize the k axes intercept points of its SRR, and the smallest standard simplex that includes the SRR. We use these results to show that the SRR grows with the increasing number of systematic columns in the generator matrices. We establish a graph-theoretic framework associating this SRR problem with fractional matchings in quasi-uniform hypergraphs. Identifying the SRR polytope is equivalent to determining a particular image of the fractional-matching polytope. We introduce a notion of Greedy Matching and show that it is sufficient to focus on these matchings to characterize the SRR rather than the entire matching polytope. With these tools, we determine the SRR of a large subset of the considered class of codes. Our results generalize previous characterizations of systematic and non-systematic MDS-coded systems, offering a unified framework for analyzing service rate regions of codes.

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arXiv2025

On the Redundancy of Function-Correcting Codes over Finite Fields

Function-correcting codes (FCCs) protect specific function evaluations of a message against errors. This condition imposes a less stringent distance requirement than classical error-correcting codes (ECCs), allowing for reduced redundancy. FCCs were introduced by Lenz et al. (2021), who also established a lower bound on the optimal redundancy for FCCs over the binary field. Here, we derive an upper bound within a logarithmic factor of this lower bound. We show that the same lower bound holds for any finite field. Moreover, we show that this bound is tight for sufficiently large fields by demonstrating that it also serves as an upper bound. Furthermore, we construct an encoding scheme that achieves this optimal redundancy. Finally, motivated by these two extreme regimes, we conjecture that our bound serves as a valid upper bound across all finite fields.

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arXiv2025

On the Service Rate Region of Reed-Muller Codes

We study the Service Rate Region of Reed-Muller codes in the context of distributed storage systems. The service rate region is a convex polytope comprising all achievable data access request rates under a given coding scheme. It represents a critical metric for evaluating system efficiency and scalability. Using the geometric properties of Reed-Muller codes, we characterize recovery sets for data objects, including their existence, uniqueness, and enumeration. This analysis reveals a connection between recovery sets and minimum-weight codewords in the dual Reed-Muller code, providing a framework for identifying those recovery sets. Leveraging these results, we derive explicit and tight bounds on the maximal achievable demand for individual data objects, thereby defining the maximal simplex within the service rate region and the smallest simplex containing it. These two provide a tight approximation of the service rate region of Reed-Muller codes.

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